Now consider integral (A.3).
(ii) We consider integral boundary conditions.
From now on, without loss of generality, we will consider the Volterra integral equation (1.3).
As an application of our results, inspired by [26], we will consider the following integral equation: x ( t ) = h ( t ) + ∫ 0 T g ( t, s ) F ( s, x ( s ) ) d s, t ∈ I = [ 0, T ]. (4.1).
First, we will consider the following triple integral which is obviously invariant under any permutations of (w_{1},w_{2},w_{3}).
For this purpose, we will consider first the following integral equation: x t)= int_{t_{0}}^{t}fbigl(x tau),x tau),ldots,x tau), taubigr),dtau+g(t), quad t in[t_{0}-delta,t_{0}+ delta], where (gin C(I)) is a given function and f is as before.
It is worth to point out that the modulation space is a better substitution to study the strongly singular integrals because there is no restriction on the index p. Here we will consider the strongly singular integrals along homogeneous curves (T_{n, beta, gamma}) on the α-modulation spaces.
In addition, we will consider the strongly singular integrals along a well-curved (Gamma (t)) in (mathbb{R}^{n}).
As an application of our results, we will consider the following Volterra type integral equation: x t)=g(t)+int_{0}^{t} Omega bigl t,s,x s bigr),ds, (19) for all (tin[0,k']), where (k'>0).
For evaluation of the integrals, we will consider two cases: Case I: x ∈ (0,a) and Case II: x ∈ (b,c).
